Probabilistic response analysis for a class of nonlinear vibro-impact oscillator with bilateral constraints under colored noise excitation

Abstract

Abstract In this paper, a new stochastic method combining the stochastic averaging of quasi-conservative and energy loss is developed to analyze the probabilistic response of a class of nonlinear vibro-impact oscillator with bilateral barriers driven by colored noise. According to the relationships between the total energy and the potential energy of impact positions of the nonlinear system, the energy loss of every movement period of the unperturbed nonlinear vibro-impact system with given total energy is analyzed in detail. Based on this analysis, an averaged It o ^ stochastic differential equation is derived through the stochastic method which combining a nonlinear transformation and the stochastic averaging of quasi-conservative, in which the averaged It o ^ equation has the typical characteristics of piecewise smooth. Next, the stationary probability density function (SPDF) is obtained by solving the averaged drift and diffusion terms of the corresponding piecewise smooth Fokker-Planck equation for the averaged It o ^ equation with the Fourier series expansion. Finally, an example is given to validate the effectiveness of the proposed method. The effects of four physical quantities of the stochastic vibro-impact system: the positions of impact, restitution coefficient, noise intensity and correlation time of the colored noise, on the probabilistic response are discussed by the SPDFs in detail, and the results are verified through the Monte Carlo simulation.